The group direct sum of a sequence 
of groups 
is the set of all sequences 
, where each 
is an element of 
, and 
is equal to the identity
element of 
for all but a finite set of indices 
. It is denoted
 |
(1)
|
and it is a group with respect to the componentwise operation derived from the operations of the groups 
.
This definition can easily be extended to any collection 
of groups, where 
is any finite or infinite set of indices.
If the additional condition on the identity elements is dropped, we get the definition of the group direct product. Hence, the two
notions coincide whenever the set of indices is finite. Thus, for any groups 
and 
,
 |
(2)
|
denote the same object.
If 
and 
are subgroups of the same additive group 
, the equality
 |
(3)
|
conventionally means that every 
has a unique decomposition 
, where 
and 
, so that 
is essentially the same as the set of all ordered pairs 
.
